AND DOUBLE THETA-FUNCTIONS. 
925 
viz., we have (j^j log Cu expressed in terms of the quotient-function , and conse¬ 
quently C u given as an exponential, the argument of which depends on the double 
du du 
D he 
integral jc— 
53. To complete the result I write the equation in the form 
d? . _ C"0 1 /D'0\2 , 1 m 'OW 7 D hi 
- 1<*°“= CO ~k ( CO ) +*( co -) y-* c% 
du 
D’O C ,; 0 
—— is = — v /Jc K, and —- is =K(K — E); hence the equation is 
C/0 go 
Aio g c„=KL -|-i j£), =kH-|-wk,|, 
or integrating twice, observing that — log Cu and log Gw, for u— 0, become =0 and 
log CO respectively, 
log C u= log C0+Gl — \ K du diiKhvdK.ii, 
which is in fact 
E 
log @(K«) = log CO-Ed- ^ 1 — y j K 2 id — k~ du | duKhvdKu, 
agreeing with Jacobi’s 
log ©w= log ©0-1—g ^1 — — ^ u '■— /cj du | 
du sn z u. 
Elliptic integrals of the third kind. 
54. We may write 
A(u + u')A(u—u!) 
1 
Vi 
AHAhi 
— ^gll 
"h 
1 
e 
1 
^5 
B (w + w')B(w—?t') 
1 
V 3 . 
W’u&td 
33 hf b—x.b—x’' 
Q(yi + u')Q,(u — u') 
1 
Vs 
GWu' 
“®fg 
c — x.c—x' 
D (m + u'yD(u—u') 
1 
x—x 
Dh'DW d—x.d—x' 
